Flat subspaces of the $SL(n,\mathbb{R})$ chiral equations

TL;DR

This paper presents a method to derive exact solutions to higher-dimensional vacuum Einstein equations by reducing them to a chiral equation involving $SL(n,\mathbb{R})$ matrices, using an ansatz with commuting matrices and generalized Laplace equations.

Contribution

The authors introduce a novel approach linking chiral equations with Einstein solutions in higher dimensions, utilizing an ansatz with commuting matrices and generalized Laplace equations.

Findings

Derived explicit solutions to the vacuum Einstein equations in higher dimensions.

Established a connection between chiral equations and Einstein metrics with Killing vectors.

Provided a framework for generating solutions using $SL(n,\mathbb{R})$ matrix parametrizations.

Abstract

In this work, we introduce a method for finding exact solutions to the vacuum Einstein field equations in higher dimensions from a given solution to the chiral equation. When considering a $n + 2$-dimensional spacetime with $n$ commutative Killing vectors, the metric tensor can take the form $\hat g = f ( \rho, \zeta ) ( d \rho^2 + d \zeta^2 ) + g_{\mu \nu} ( \rho, \zeta ) d x^\mu d x^\nu$. Then, the Einstein field equations in vacuum reduce to a chiral equation, $( \rho g_{, z} g ^{-1} )_{, \bar z} + ( \rho g_{, \bar z} g ^{-1} )_{, z} = 0$, and two differential equations, $( \ln f \rho ^{1-1/n} )_{, Z} = \frac{\rho}{2} \operatorname{tr} ( g_{, _Z} g^{-1} )^2$, where $g \in SL( n, \mathbb{R} )$ is the normalized matrix representation of $g_{\mu \nu}$, $z = \rho + i \zeta$ and $Z = z, \bar z$. We use the ansatz $g = g ( \xi^a )$, where the parameters $\xi^a$ depend on $z$ and $\bar z$ and satisfy a generalized Laplace equation, $( \rho \xi^a _{, z} )_{, \bar z} + ( \rho \xi^a _{, \bar z} )_{, z} = 0$. The chiral equation to the Killing equation, $A_{a , \xi^b} + A_{b , \xi^a} = 0$, where $A_a = g_{, \xi^a} g^{-1}$. Furthermore, we assume that the matrices $A_a$ commute with each other; in this way, they fulfill the Killing equation.